DOCSLIB.ORG

Regression, Causality and Identification Issues Kamiljon T

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Regression, Causality and Identification Issues Kamiljon T Regression, Causality and Identification Issues Kamiljon T. Akramov, Ph.D. IFPRI, Washington, DC, USA Training Course on Applied Econometric Analysis June 4, 2015, WIUT, Tashkent, Uzbekistan Standard OLS Model: Summary • Consider a simple regression model Y = β0 + β1X1 + β2X2 + u • It is assumed to satisfy the following four conditions • The model is linear in the parameters β • n observations are drawn randomly from populations of interest • E(u|X1, X2)=0, i.e., no endogeniety in the true model • No perfect collinearity • These four assumptions guarantee that the OLS estimates of parameters will be unbiased Standard OLS model: Summary (cont.) • Standard OLS model provides an estimate of the effect on Y of arbitrary changes in independent variables (∆X) • It resolves the problem of omitted variable bias, if an omitted variable can be measured and included • It can handle certain nonlinear relations (effects that vary with the X’s) • Still, standard OLS might yield a biased estimator of the true causal effect Motivation • The most challenging empirical questions in economics involve causal-effect relationships: • Will mandatory health insurance really make people healthier? • How does an additional year of education change earnings? • What is the effect of farm size on agricultural productivity? • How does agricultural diversity impact nutritional outcomes? Does Chocolate Consumption enhance cognitive function? Source: Messerli (2012) Regression analysis • “Essentially, all (statistical) models are wrong, but some are useful” George E. P. Box (1987) • All regression (statistical) models are description of real world phenomenon using mathematical concepts, i.e., they are just simplifications of reality • Regression analysis can be very useful if it is carefully designed • In accordance with current good practice guidelines, and • A thorough understanding of the limitations of the methods used • If not, it can be not only inaccurate but also potentially damaging by misleading policymakers, practitioners and public • Example 1: impact of hormone replacement therapy on the risk of coronary heart diseases (Hully et al. 1998; Velickovic 2015) • Example 2: Relationship between levels of government debt and rates of economic growth (Reinhart & Rogoff controversy) Regression and causality • The aim of standard regression analysis is to infer parameters of a distribution from samples drawn of that distribution • With the help of such parameters, one can: • Infer association among variables • Estimate the likelihood of past and future events • As well as update the likelihood of events in light of new evidence or new measurement • Causal analysis goes one step further: • Its aim is to infer aspects of the data generation process • With the help of such aspects, one can deduce not only the likelihood of events under static conditions, but also the dynamics of events under changing conditions 7 Causal analysis: schooling and earnings • Causal relationship between schooling and earnings tells us what people would earn, on average, if we could either • Change their schooling in a perfectly controlling environment or • Change their schooling randomly so that those with different levels of schooling would otherwise comparable • Conditional independence assumption (CIA) requires that we must hold a variety of control variables fixed for causal inferences to be valid • Selection on observables • Covariates to be fixed are assumed to be known and observed Causal analysis: schooling and earnings • Assume schooling is a binary decision, • Two potential earnings variables Y1i if ci =1 Outcome = Y0i if ci = 0 • We would like to know the difference between , which is causal effect of schooling on individual 1 2 Yi = Y0i + (Y1i −Y0i )ci Causal analysis: schooling and earnings • Comparison of average earning conditional on schooling status is formally linked to the average causal effect, i.e., Observed difference=Average treatment effect on treated + Selection bias E[Yi | Ci =1]− E[Yi | Ci = 0]= E[Y1i −Y0i | Ci =1] + E[Y0i | Ci =1]− E[Y0i | Ci = 0] • If selection bias is positive, the naïve comparison of earnings exaggerates the benefits of schooling • CIA asserts that conditional on observed characteristics selection bias disappears and comparisons of average earnings across schooling levels have a causal interpretation Fundamental problem of causal inference • It is impossible to observe the value of Y1i and Y0i on the same individual and, therefore, it is impossible to directly observe the effect of schooling on earnings • Another way to express this problem is to say that we cannot infer the effect of schooling because we do not have the counterfactual evidence, i.e., what would have happened in the absence of schooling • Given that the causal effect for a single individual cannot be observed, we aim to identify the average causal effect for the entire population or for sub-populations Fundamental problem of causal inference: solution • The econometric solution replaces the impossible-to-observe causal effect of treatment on a specific unit with the possible-to-estimate average causal effect of treatment over a population of units • Although E(Y1i) and E(Y0i) cannot both be calculated, they can be estimated. • Most econometrics methods attempt to construct from observational data consistent estimates of Y1i and Y0i Causal analysis: additional issues • In most circumstances, there is simply no information available on how those in the control group would have reacted if they had received the treatment instead • This is the basis for an important insight into another potential biase of standard regression analysis – treatment heterogeneity • Thus, two sources of biases need to be eliminated from estimates of causal effects from observational studies 1. Selection Bias: Baseline difference 2. Treatment Heterogeneity • Most of the methods available only deal with selection bias, simply assuming that the treatment effect is constant in the population or by redefining the parameter of interest in the population Macro example • What explains income differences across countries? • Hypothesis: the quality of institutions explains the variation in per capita income across countries • How would you establish causal link between institutions and income? • Higher levels of economic development may cause higher levels of institutional quality • Unobserved variable may jointly determine both high levels of institutional quality and high levels of income Internal Validity • Recall our definition of internal validity: • An econometric analysis is internally valid if the statistical inferences about causal effects are valid for the population being studied • We know that internal validity hinges on two things: • The estimator of the causal effect should be consistent (unbiased would be nice too, but it’s not always feasible) • Hypothesis tests should have the desired significance level (i.e. you should be using the correct standard errors) Threats to Internal Validity • Omitted variable bias • Model misspecification or wrong functional form • Measurement error • Selection bias • Simultaneous causality bias • All of these imply that E(ui|X1,X2) ≠ 0 Omitted Variable Bias • The bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variable bias • For omitted variable bias to occur, the omitted factor “Z” must be: • a determinant of Y; and • correlated with the regressor X but unobserved, so cannot be included in the regression • Both conditions must hold for the omission of Z to result in omitted variable bias Omitted Variable Bias Formula • Regression of wages on schooling = α + ρ + + where α, ρ, and are population regression coefficients and is a regression residual that is uncorrelated with all regressor • What are the consequences of leaving ability out of regression? • OVB formula ( , ) ( ) = ρ + Where is the vector of coefficients from regressions of the elements of and Potential Solutions to Omitted Variable Bias • If the variable can be measured and observed, include it as a regressor in multiple regression • But be careful about bad control variables • Possibly, use panel data in which each entity (individual) is observed more than once • Fixed effects to control for unobserved heterogeneity • If the variable cannot be measured, use instrumental variables regression • Run a randomized controlled experiment OVB Example: estimates of the returns to education for men in the NLSY Controls (1) (2) (3) (4) (5) None Age dummies Col. (2) and Col. (4) and Col. (4) and additional AFQT score occupation control variables dummies (mother’s and father’s years of schooling, and dummies for race and census region) 0.132 0.131 0.114 0.087 0.066 (0.007) (0.007) (0.007) (0.009) (0.010) Table reports the coefficient on years of schooling in a regression of log wages on years of schooling and the indicated controls. Source: Angrist and Pischke (2009). Misspecification or Wrong Functional Form • Arises if the functional form is incorrect • If an interaction term is incorrectly omitted, then inferences on causal effects will be biased • Variable transformations (logarithms) • Discrete dependent variables • For example, the effect of dietary diversity on nutritional outcomes may depend on children’s age • Other examples? Measurement Error • In reality, economic data often have measurement error • Data entry errors in administrative data • Recollection errors in surveys • when did you start your current job? • Ambiguous questions problems
Load more

Recommended publications
  • A Difference-Making Account of Causation1
    A difference-making account of causation1 Wolfgang Pietsch2, Munich Center for Technology in Society, Technische Universität München, Arcisstr. 21, 80333 München, Germany A difference-making account of causality is proposed that is based on a counterfactual definition, but differs from traditional counterfactual approaches to causation in a number of crucial respects: (i) it introduces a notion of causal irrelevance; (ii) it evaluates the truth-value of counterfactual statements in terms of difference-making; (iii) it renders causal statements background-dependent. On the basis of the fundamental notions ‘causal relevance’ and ‘causal irrelevance’, further causal concepts are defined including causal factors, alternative causes, and importantly inus-conditions. Problems and advantages of the proposed account are discussed. Finally, it is shown how the account can shed new light on three classic problems in epistemology, the problem of induction, the logic of analogy, and the framing of eliminative induction. 1. Introduction ......................................................................................................................................................... 2 2. The difference-making account ........................................................................................................................... 3 2a. Causal relevance and causal irrelevance ....................................................................................................... 4 2b. A difference-making account of causal counterfactuals
    [Show full text]
  • A Visual Analytics Approach for Exploratory Causal Analysis: Exploration, Validation, and Applications
    A Visual Analytics Approach for Exploratory Causal Analysis: Exploration, Validation, and Applications Xiao Xie, Fan Du, and Yingcai Wu Fig. 1. The user interface of Causality Explorer demonstrated with a real-world audiology dataset that consists of 200 rows and 24 dimensions [18]. (a) A scalable causal graph layout that can handle high-dimensional data. (b) Histograms of all dimensions for comparative analyses of the distributions. (c) Clicking on a histogram will display the detailed data in the table view. (b) and (c) are coordinated to support what-if analyses. In the causal graph, each node is represented by a pie chart (d) and the causal direction (e) is from the upper node (cause) to the lower node (result). The thickness of a link encodes the uncertainty (f). Nodes without descendants are placed on the left side of each layer to improve readability (g). Users can double-click on a node to show its causality subgraph (h). Abstract—Using causal relations to guide decision making has become an essential analytical task across various domains, from marketing and medicine to education and social science. While powerful statistical models have been developed for inferring causal relations from data, domain practitioners still lack effective visual interface for interpreting the causal relations and applying them in their decision-making process. Through interview studies with domain experts, we characterize their current decision-making workflows, challenges, and needs. Through an iterative design process, we developed a visualization tool that allows analysts to explore, validate, arXiv:2009.02458v1 [cs.HC] 5 Sep 2020 and apply causal relations in real-world decision-making scenarios.
    [Show full text]
  • Causal Inference for Process Understanding in Earth Sciences
    Causal inference for process understanding in Earth sciences Adam Massmann,* Pierre Gentine, Jakob Runge May 4, 2021 Abstract There is growing interest in the study of causal methods in the Earth sciences. However, most applications have focused on causal discovery, i.e. inferring the causal relationships and causal structure from data. This paper instead examines causality through the lens of causal inference and how expert-defined causal graphs, a fundamental from causal theory, can be used to clarify assumptions, identify tractable problems, and aid interpretation of results and their causality in Earth science research. We apply causal theory to generic graphs of the Earth sys- tem to identify where causal inference may be most tractable and useful to address problems in Earth science, and avoid potentially incorrect conclusions. Specifically, causal inference may be useful when: (1) the effect of interest is only causally affected by the observed por- tion of the state space; or: (2) the cause of interest can be assumed to be independent of the evolution of the system’s state; or: (3) the state space of the system is reconstructable from lagged observations of the system. However, we also highlight through examples that causal graphs can be used to explicitly define and communicate assumptions and hypotheses, and help to structure analyses, even if causal inference is ultimately challenging given the data availability, limitations and uncertainties. Note: We will update this manuscript as our understanding of causality’s role in Earth sci- ence research evolves. Comments, feedback, and edits are enthusiastically encouraged, and we arXiv:2105.00912v1 [physics.ao-ph] 3 May 2021 will add acknowledgments and/or coauthors as we receive community contributions.
    [Show full text]
  • Equations Are Effects: Using Causal Contrasts to Support Algebra Learning
    Equations are Effects: Using Causal Contrasts to Support Algebra Learning Jessica M. Walker ([email protected]) Patricia W. Cheng ([email protected]) James W. Stigler ([email protected]) Department of Psychology, University of California, Los Angeles, CA 90095-1563 USA Abstract disappointed to discover that your digital video recorder (DVR) failed to record a special television show last night U.S. students consistently score poorly on international mathematics assessments. One reason is their tendency to (your goal). You would probably think back to occasions on approach mathematics learning by memorizing steps in a which your DVR successfully recorded, and compare them solution procedure, without understanding the purpose of to the failed attempt. If a presetting to record a regular show each step. As a result, students are often unable to flexibly is the only feature that differs between your failed and transfer their knowledge to novel problems. Whereas successful attempts, you would readily determine the cause mathematics is traditionally taught using explicit instruction of the failure -- the presetting interfered with your new to convey analytic knowledge, here we propose the causal contrast approach, an instructional method that recruits an setting. This process enables you to discover a new causal implicit empirical-learning process to help students discover relation and better understand how your DVR works. the reasons underlying mathematical procedures. For a topic Causal induction is the process whereby we come to in high-school algebra, we tested the causal contrast approach know how the empirical world works; it is what allows us to against an enhanced traditional approach, controlling for predict, diagnose, and intervene on the world to achieve an conceptual information conveyed, feedback, and practice.
    [Show full text]
  • Judea's Reply
    Causal Analysis in Theory and Practice On Mediation, counterfactuals and manipulations Filed under: Discussion, Opinion — moderator @ 9:00 pm , 05/03/2010 Judea Pearl's reply: 1. As to the which counterfactuals are "well defined", my position is that counterfactuals attain their "definition" from the laws of physics and, therefore, they are "well defined" before one even contemplates any specific intervention. Newton concluded that tides are DUE to lunar attraction without thinking about manipulating the moon's position; he merely envisioned how water would react to gravitaional force in general. In fact, counterfactuals (e.g., f=ma) earn their usefulness precisely because they are not tide to specific manipulation, but can serve a vast variety of future inteventions, whose details we do not know in advance; it is the duty of the intervenor to make precise how each anticipated manipulation fits into our store of counterfactual knowledge, also known as "scientific theories". 2. Regarding identifiability of mediation, I have two comments to make; ' one related to your Minimal Causal Models (MCM) and one related to the role of structural equations models (SEM) as the logical basis of counterfactual analysis. In my previous posting in this discussion I have falsely assumed that MCM is identical to the one I called "Causal Bayesian Network" (CBM) in Causality Chapter 1, which is the midlevel in the three-tier hierarchy: probability-intervention-counterfactuals? A Causal Bayesian Network is defined by three conditions (page 24), each invoking probabilities
    [Show full text]
  • Causal Analysis of COVID-19 Observational Data in German Districts Reveals Effects of Mobility, Awareness, and Temperature
    medRxiv preprint doi: https://doi.org/10.1101/2020.07.15.20154476; this version posted July 23, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC 4.0 International license . Causal analysis of COVID-19 observational data in German districts reveals effects of mobility, awareness, and temperature Edgar Steigera, Tobias Mußgnuga, Lars Eric Krolla aCentral Research Institute of Ambulatory Health Care in Germany (Zi), Salzufer 8, D-10587 Berlin, Germany Abstract Mobility, awareness, and weather are suspected to be causal drivers for new cases of COVID-19 infection. Correcting for possible confounders, we estimated their causal effects on reported case numbers. To this end, we used a directed acyclic graph (DAG) as a graphical representation of the hypothesized causal effects of the aforementioned determinants on new reported cases of COVID-19. Based on this, we computed valid adjustment sets of the possible confounding factors. We collected data for Germany from publicly available sources (e.g. Robert Koch Institute, Germany’s National Meteorological Service, Google) for 401 German districts over the period of 15 February to 8 July 2020, and estimated total causal effects based on our DAG analysis by negative binomial regression. Our analysis revealed favorable causal effects of increasing temperature, increased public mobility for essential shopping (grocery and pharmacy), and awareness measured by COVID-19 burden, all of them reducing the outcome of newly reported COVID-19 cases. Conversely, we saw adverse effects of public mobility in retail and recreational areas, awareness measured by searches for “corona” in Google, and higher rainfall, leading to an increase in new COVID-19 cases.
    [Show full text]
  • 12.4 Methods of Causal Analysis 519
    M12_COPI1396_13_SE_C12.QXD 11/13/07 9:39 AM Page 519 12.4 Methods of Causal Analysis 519 therefore unlikely even to notice, the negative or disconfirming instances that might otherwise be found. For this reason, despite their fruitfulness and value in suggesting causal laws, inductions by simple enumeration are not at all suitable for testing causal laws. Yet such testing is essential; to accomplish it we must rely upon other types of inductive arguments—and to these we turn now. 12.4 Methods of Causal Analysis The classic formulation of the methods central to all induction were given in the nineteenth century by John Stuart Mill (in A System of Logic, 1843). His systematic account of these methods has led logicians to refer to them as Mill’s methods of inductive inference. The techniques themselves—five are commonly distinguished—were certainly not invented by him, nor should they be thought of as merely a product of nineteenth-century thought. On the contrary, these are universal tools of scientific investigation. The names Mill gave to them are still in use, as are Mill’s precise formulations of what he called the “canons of induction.” These techniques of investigation are per- manently useful. Present-day accounts of discoveries in the biological, social, and physical sciences commonly report the methodology used as one or another variant (or combination) of these five techniques of inductive infer- ence called Mill’s methods: They are 1. The method of agreement 2. The method of difference 3. The joint method of agreement and difference *Induction by simple enumeration can mislead even the learned.
    [Show full text]
  • Chapter 17 Social Networks and Causal Inference
    Chapter 17 Social Networks and Causal Inference Tyler J. VanderWeele and Weihua An Abstract This chapter reviews theoretical developments and empirical studies related to causal inference on social networks from both experimental and observational studies. Discussion is given to the effect of experimental interventions on outcomes and behaviors and how these effects relate to the presence of social ties, the position of individuals within the network, and the underlying structure and properties of the network. The effects of such experimental interventions on changing the network structure itself and potential feedback between behaviors and network changes are also discussed. With observational data, correlations in behavior or outcomes between individuals with network ties may be due to social influence, homophily, or environmental confounding. With cross-sectional data these three sources of correlation cannot be distinguished. Methods employing longitudinal observational data that can help distinguish between social influence, homophily, and environmental confounding are described, along with their limitations. Proposals are made regarding future research directions and methodological developments that would help put causal inference on social networks on a firmer theoretical footing. Introduction Although the literature on social networks has grown dramatically in recent years (see Goldenberg et al. 2009;An2011a, for reviews), formal theory for inferring causation on social networks is arguably still in its infancy. As described elsewhere in this handbook, much of the recent theoretical development within causal inference has been within a “counterfactual-” or “potential outcomes-” based perspective (Rubin 1974; see also Chap. 5 by Mahoney et al., this volume), and this approach typically employs a “no interference” assumption (Cox 1958; Rubin 1980; see also Chap.
    [Show full text]
  • Confounding and Control
    Demographic Research a free, expedited, online journal of peer-reviewed research and commentary in the population sciences published by the Max Planck Institute for Demographic Research Konrad-Zuse Str. 1, D-18057 Rostock · GERMANY www.demographic-research.org DEMOGRAPHIC RESEARCH VOLUME 16, ARTICLE 4, PAGES 97-120 PUBLISHED 06 FEBRUARY 2007 http://www.demographic-research.org/Volumes/Vol16/4/ DOI: 10.4054/DemRes.2007.16.4 Research Article Confounding and control Guillaume Wunsch © 2007 Wunsch This open-access work is published under the terms of the Creative Commons Attribution NonCommercial License 2.0 Germany, which permits use, reproduction & distribution in any medium for non-commercial purposes, provided the original author(s) and source are given credit. See http:// creativecommons.org/licenses/by-nc/2.0/de/ Table of Contents 1 Introduction 98 2 Confounding 100 2.1 What is confounding? 100 2.2 A confounder as a common cause 101 3 Control 104 3.1 Controlling ex ante: randomisation in prospective studies 104 3.2 Treatment selection in non-experimental studies 106 3.3 Controlling ex post in non-experimental studies 107 3.4 Standardisation 108 3.5 Multilevel modelling 109 3.6 To control … ? 110 3.7 Controlling for a latent confounder 110 3.8 … or not to control? 112 3.9 Common causes and competing risks 114 4 Conclusions 115 5 Acknowledgments 116 References 117 Demographic Research: Volume 16, Article 4 research article Confounding and control Guillaume Wunsch 1 Abstract This paper deals both with the issues of confounding and of control, as the definition of a confounding factor is far from universal and there exist different methodological approaches, ex ante and ex post, for controlling for a confounding factor.
    [Show full text]
  • Causal Laws and Causal Analysis
    Esc'r.r THE CAUSAL VERSION OF THE MODEL 87 uncommon for philosophers to represent causal laws as having special explanatory force. Thus C. Ducasse, after defining IV J. explanation in terms of subsumption under a 'law of con- nection', and having added that mere'law CAUSAL LAWS AND CAUSAL ANALYSIS a of correlation' will not do, goes on to say that laws of the former sort are either causal or logical.' And many contemporary philosophers of t. The Causal Versionof the Model science, with quantum physics in mind, would agree with o far I have said very little about specifically causal Mr. A. P. Ushenko that causal laws alone have "explanatory explanation.In ChapterII, althoughcausal language was virtue".2 not avoidedaltogether, our concernwas chieflytotestthe No doubt many of those who have phrased the covering law covering law claims with respect to explanations which were claim in terms of specifically causal laws have used the term complete in a special sense, and which would not necessarily, 'causal' carelessly.Some have meant no more than 'empirical or even naturally, be formulated in causal terms. In Chapter laws', by contrast with, say, general principles of logic. Others III, too, no attempt was made to contrast explanations given have probably had in mind a distinction within the class by making reference to causal laws with explanations of other of empirical laws, between mere 'probability hypotheses' or kinds. But some defenders of the model have stated their statistical generalizations and genuinely universal laws-for claims explicitly in terms of covering causal laws, as if sub- causal laws are often held to set forth invariable connexions.
    [Show full text]
  • Ch. 4: Statistical Research Designs for Causal Inference∗
    Ch. 4: Statistical research designs for causal inference∗ Fabrizio Gilardiy January 24, 2012 1 Introduction In Chapter 3 we have discussed the different ways in which the social sciences conceptualize causation and we have argued that there is no single way in which causal relationships can be defined and analyzed empirically. In this chapter, we focus on a specific set of approaches to constructing research designs for causal analysis, namely, one based on the potential outcomes framework developed in statistics. As discussed in Chapter 3, this perspective is both proba- bilistic and counterfactual. It is probabilistic because it does not assume that the presence of a given cause leads invariably to a given effect, while it is counterfactual because it involves the comparison of actual configurations with hypothetical alternatives that are not observed in reality. In essence, this approach underscores the necessity to rely on comparable groups in order to achieve valid causal inferences. An important implication is that the design of a study is of paramount importance. The way in which the data are produced is the critical step of the research, while the actual data analysis, while obviously important, plays a secondary role. However, a convincing design requires research questions to be broken down to manageable pieces. Thus, the big tradeoff in this perspective is between reliable inferences on very specific causal relationships on the one hand, and their broader context and complexity (and, possibly, theoretical relevance) on the other hand. The chapter first distinguishes between two general perspectives on causality, namely, one that puts the causes of effects in the foreground, and another that is more interested in the effects of causes.
    [Show full text]
  • Causal Inference in Statistics: an Overview∗†‡
    Statistics Surveys Vol. 3 (2009) 96–146 ISSN: 1935-7516 DOI: 10.1214/09-SS057 Causal inference in statistics: An overview∗†‡ Judea Pearl Computer Science Department University of California, Los Angeles, CA 90095 USA e-mail: [email protected] Abstract: This review presents empirical researcherswith recent advances in causal inference, and stresses the paradigmatic shifts that must be un- dertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that un- derly all causal inferences, the languages used in formulating those assump- tions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and unifies other approaches to causation, and provides a coher- ent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the effects of potential interven- tions, (also called “causal effects” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret,” “attri- bution” or “causes of effects”) and (3) queries about direct and indirect effects (also known as “mediation”). Finally, the paper defines the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both. Keywords and phrases: Structuralequation models, confounding,graph- ical methods, counterfactuals, causal effects, potential-outcome, mediation, policy evaluation, causes of effects.
    [Show full text]